How Many Triangles are There??

Consider only one tier of triangles as shown. We note that a triangle is formed by the top point $0$ with any two of the bottom four points. Therefore, the total of number of triangles formed in one tier is $\dbinom 42 = 6$ . Since there are four tiers, the total number of triangles formed is $6 \times 4 = \boxed{24}$ .

The triangles in the figure follow the following rule

• the common vertex of all triangle is the upper one.
• the other two point must be in the same line except the adjacent line of the point.

There are $4$ points in every line. So choosing $2$ point = ${4 \choose 2}$ = $6$ and as each line contains $4$ point $6$ $\times$ $4$ $=$ $\boxed{24}$